Equivalence Classes Math Example 1

Solution

1. 1
   Reflexive: $a \equiv a \pmod{3}$ (since $3 \mid 0$). True.
2. 2
   Symmetric: if $3 \mid (a-b)$, then $3 \mid (b-a)$. True.
3. 3
   Transitive: if $3 \mid (a-b)$ and $3 \mid (b-c)$, then $3 \mid (a-c)$ (by addition). True.
4. 4
   Equivalence classes: $[0]=\{\ldots,-3,0,3,6,\ldots\}$, $[1]=\{\ldots,-2,1,4,7,\ldots\}$, $[2]=\{\ldots,-1,2,5,8,\ldots\}$. These three classes partition $\mathbb{Z}$.